A334618 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 4, where the m-th block starts with m, m >= 1, and the first element of column k is in the row that is the k-th hexagonal number (A000384).

1, 2, 3, 4, 5, 6, 1, 7, 5, 8, 2, 9, 6, 10, 3, 11, 7, 12, 4, 13, 8, 14, 5, 15, 9, 1, 16, 6, 5, 17, 10, 9, 18, 7, 2, 19, 11, 6, 20, 8, 10, 21, 12, 3, 22, 9, 7, 23, 13, 11, 24, 10, 4, 25, 14, 8, 26, 11, 12, 27, 15, 5, 28, 12, 9, 1, 29, 16, 13, 5, 30, 13, 6, 9, 31, 17, 10, 13, 32, 14, 14, 2

Offset: 1

Author

Omar E. Pol, Dec 18 2020

Keywords

• nonn
• tabf

Comments

This triangle can be interpreted as a table of partitions into consecutive parts that differ by 4 (see the Example section).

Also, every triangle of this family has the property that starting from row n the sum of k positive and consecutive terms in the column k is equal to n.

Examples

			Triangle begins (rows 1..28):
   1;
   2;
   3;
   4;
   5;
   6,  1;
   7,  5;
   8,  2;
   9,  6;
  10,  3;
  11,  7;
  12,  4;
  13,  8;
  14,  5;
  15,  9,  1;
  16,  6,  5;
  17, 10,  9;
  18,  7,  2;
  19, 11,  6;
  20,  8, 10;
  21, 12,  3;
  22,  9,  7;
  23, 13, 11;
  24, 10,  4;
  25, 14,  8;
  26, 11, 12;
  27, 15,  5;
  28, 12,  9,  1;
...
Figures A..H show the location (in the columns of the table) of the partitions of n = 1..8 (respectively) into consecutive parts that differ by 4:
.   -----------------------------------------------------------
Fig:   A     B     C     D     E       F        G        H
.   -----------------------------------------------------------
. n:   1     2     3     4     5       6        7        8
Row -----------------------------------------------------------
1   | [1];|  1; |  1; |  1; |  1; |  1;     |  1;   |  1;     |
2   |     | [2];|  2; |  2; |  2; |  2;     |  2;   |  2;     |
3   |     |     | [3];|  3; |  3; |  3;     |  3;   |  3;     |
4   |     |     |     | [4];|  4; |  4;     |  4;   |  4;     |
5   |     |     |     |     | [5];|  5;     |  5;   |  5;     |
6   |     |     |     |     |     | [6],[1];|  6, 1;|  6,  1; |
7   |     |     |     |     |     |     [5];| [7],5;|  7,  5; |
8   |     |     |     |     |     |         |       | [8],[2];|
9   |     |     |     |     |     |         |       |  9, [6];|
.   -----------------------------------------------------------
Figure H: for n = 8 the partitions of 8 into consecutive parts that differ by 4 (but with the parts in increasing order) are [8] and [2, 6]. These partitions have one part and two parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 2 of this table, starting at the row 8.
.
Illustration of initial terms arranged into a triangular structure:
.                                                           _
.                                                         _|1|
.                                                       _|2  |
.                                                     _|3    |
.                                                   _|4      |
.                                                 _|5       _|
.                                               _|6        |1|
.                                             _|7         _|5|
.                                           _|8          |2  |
.                                         _|9           _|6  |
.                                       _|10           |3    |
.                                     _|11            _|7    |
.                                   _|12             |4      |
.                                 _|13              _|8      |
.                               _|14               |5       _|
.                             _|15                _|9      |1|
.                           _|16                 |6        |5|
.                         _|17                  _|10      _|9|
.                       _|18                   |7        |2  |
.                     _|19                    _|11       |6  |
.                   _|20                     |8         _|10 |
.                 _|21                      _|12       |3    |
.               _|22                       |9          |7    |
.              |23                         |13         |11   |
...
The number of horizontal line segments in the n-th row of the diagram equals A334461(n), the number of partitions of n into consecutive parts that differ by 4.
		

Crossrefs

Tables of the same family where the consecutive parts differ by d are A010766 (d=0), A286001 (d=1), A332266 (d=2), A334945 (d=3), this sequence (d=4).

Cf. A000384, A327262, A334460, A334461, A334462, A334464.